My experience, and I heard the same from a couple of colleagues that manage companies in the test automation domain, is that the best approach is to start small and "look under the street-lamp".
The logic behind it is that there is simply too much to learn and try in order to achieve good test automation (or checking, whatever) and while you are doing that the product is being developed and changed, technologies change, and the test frameworks themselves change.
So what I suggest, and this is not the BEST approach but simply a feasible one that has a good chance not to waste a lot of un-successful trials and resources, is to do a very short investigation like the other answers suggest, but shorter, choose one way and implement a very very basic test.
But, and there is a but, make sure that this test is reliable, can be extended easily, can be understood by others and if I haven't emphasized enough it should be super stable and reliable.
You will get a simple smoke test, but on the way you will also learn a lot for the next round of automation.